#808

Soup Servings

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math · dynamic programming · probability and statistics

Description

You have two soups, A and B, each starting with n mL. On every turn, one of the following four serving operations is chosen at random, each with probability 0.25 independent of all previous turns:

- pour 100 mL from type A and 0 mL from type B

- pour 75 mL from type A and 25 mL from type B

- pour 50 mL from type A and 50 mL from type B

- pour 25 mL from type A and 75 mL from type B

Note:

- There is no operation that pours 0 mL from A and 100 mL from B.

- The amounts from A and B are poured simultaneously during the turn.

- If an operation asks you to pour more than you have left of a soup, pour all that remains of that soup.

The process stops immediately after any turn in which one of the soups is used up.

Return the probability that A is used up before B, plus half the probability that both soups are used up in the same turn. Answers within 10^-5 of the actual answer will be accepted.

Example 1:

Input: n = 50

Output: 0.62500

Explanation:

If we perform either of the first two serving operations, soup A will become empty first.

If we perform the third operation, A and B will become empty at the same time.

If we perform the fourth operation, B will become empty first.

So the total probability of A becoming empty first plus half the probability that A and B become empty at the same time, is 0.25 * (1 + 1 + 0.5 + 0) = 0.625.

Example 2:

Input: n = 100

Output: 0.71875

Explanation:

If we perform the first serving operation, soup A will become empty first.

If we perform the second serving operations, A will become empty on performing operation [1, 2, 3], and both A and B become empty on performing operation 4.

If we perform the third operation, A will become empty on performing operation [1, 2], and both A and B become empty on performing operation 3.

If we perform the fourth operation, A will become empty on performing operation 1, and both A and B become empty on performing operation 2.

So the total probability of A becoming empty first plus half the probability that A and B become empty at the same time, is 0.71875.

Solution